## Evolution in close binaries

12.1 I Close binaries

If the two stars in a binary system are very close to each other, each has the effect of altering the structure of the other star. When this occurs we call the system a close binary system. The surface of a star can be distorted by the stronger gravitational force that the companion exerts on the near side than on the far side. Remember, we said that any effect that depends on variations in the gravitational force from one position to another is called a tidal effect. (A similar situation applies as the Sun and Moon distort the Earth's ocean surface, raising the tides.)

The distortion of stars results in internal dissipation of energy. As a star rotates, different material is incorporated in the bulge. Different layers of material rub against each other, in a fluid friction. This lost energy has to come from somewhere. It comes from both the orbital energy and the rotational energy of the star. As a result, eventually the orbits circularize and the two stars always keep the same sides towards each other. This is the lowest energy arrangement for the system (see Problem 12.1). We say that the spins are synchronized. (The Moon's spin and orbital motion around the Earth are synchronized, and the Moon keeps the same side towards the Earth.)

In certain situations, it is possible for material from one star to be pulled off the surface onto the other star. To see how this can happen, we look at a binary system from a coordinate system rotating with the same period as that of the orbits. If we look at the energy of a particle in this system, the rotation of the coordinate system introduces a term in addition to the gravitational potential. This term is equal to J2/2mr2, where J, m and r are the angular momentum, mass and distance, respectively, from the origin of some particle. (We can think of it as the term in the potential energy which gives rise to the pseudo "centrifugal force" in the rotating system.) When we add this term to the gravitational potential, we have an effective potential that can be used to describe the motions of particles. We can draw surfaces of constant effective potential, as in Fig. 12.1. The effective force (gravity plus "centrifugal") at any point on one of these surface is perpendicular to the surface. (This is analogous to contour maps of gravitational potential - elevation - on Earth. The gravitational force is perpendicular to the contour lines, and you don't have to do any work to move along an equipotential line.)

There are five points, called Lagrangian points, where the effective gravitational force is zero. These points are designated L1, L2, L3, L4, L5. (Note that the L5 Society wants to place a space station at the L5 point for the Earth/Moon system.) The point L1 lies between the two stars, at the intersection point of the "figure eight" shaped surface. L1 is the dividing point between material being attracted to one star or the other. The two sides of the figure eight are called Roche lobes (Fig. 12.2).

The equipotential surfaces in a fluid must be surfaces of constant pressure. If they were not, there would be pressure differences forcing fluid along the surface, and these forces could not be balanced by any gravitational forces,

Surfaces of constant effective potential

Surfaces of constant effective potential

Equatorial plane

Surfaces of constant effective potential.The effective potential is the true gravitational potential plus that resulting from the centrifugal force in the coordinate system rotating with the orbiting system. (a) We look from above and see the intersection of surfaces of constant potential and the plane of the orbit.The heavy 'figure eight' is the intersection of the Roche lobe and the plane of the orbit. (b) A three-dimensional view of the Roche lobe. (In this case, M| ^ M2.)

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